Diophantine approximation in parameter spaces of the dynamical system of beta-transformations

Time :

2014-07-16 (Wed) 14:30 - 15:30

Place :

Seminar Room 722, Institute of Mathematics (NTU Campus)

Abstract:

We consider the distribution of the orbits of the number 1 under the -transformations as varies. Mainly, the size of the set of for which a given point can be well approximated by the orbit of 1 is measured by its Hausdorff dimension. The dimension of the following set
E = , for infinitely many n
is determined, where is a given point in and is a sequence of integers tending to infinity as . For the proof of this result, the notion of the recurrence time of a word in symbolic space is introduced to characterize the lengths and the distribution of cylinders (the set of with a common prefix in the expansion of 1) in the parameter space .